Tuesday, November 13, 2012

Marbles and Monty Hall

A collection of marbles within the permanent collection of The Children's Museum of Indianapolis.

I was confronted with this problem by a kid about 1/4 my age.  He said, "imagine I have a bag with three marbles in it; two white and one black.  If you choose the black marble you win.  You reach into the bag and pull out a marble, but don't look at it.  There are now two marbles in the bag and one in your hand.  I now pull a white marble out of the bag and show it to you.  This leaves only one marble in your hand and one in the bag.  For best probability of picking the black marble, do you keep the one in your hand or pick the one in the bag?"  This is called the Monty Hall problem because it is similar to a dilemma faced by contestants on the TV gameshow, The Price Is Right (with host Monty Hall) every week.

The somewhat counterintuitive answer is that it actually is statistically better to go for the marble in the bag instead of the one in your hand.  It took about 45 minutes for this *$#! young fellow to convince me of this.

Here's why it's true:
When you take the marble from the bag, you have a 1 in 3 chance of choosing the black marble.  That means there is a better chance that it is in the bag (2 in 3 chances) than in your hand.  So far so good.  Well, if this little know-it-all kid, shows you one of the marbles in the bag that ISN'T the black one, there is still a 2 in 3 chance that the black marble is in the bag, but now all that chance is tied up in the one marble that remains in the bag... so you should take it rather than the 1 in 3 chance marble you are holding in your hand.

The video above explains the problem rather nicely, so if my explanation didn't do it for you, try the video.  It might be worth millions the next time you are on a game show.  Just remember to send me my cut of your winnings.